210 DR THOMAS MUIR ON 



for the purpose, as before, of separating out the element concerned, thus obtaining 



d a „ \«i « 2 ■ • • a h-J W+i ■ ■ • a «-i a J • 



In other words, the differential-quotient of K with respect to a h is obtained mechanically 

 by taking K, deleting b h . lt a h , b h , and inserting the brackets )( instead. 



When one of the end elements is used as independent variable the result is of course 

 simpler : thus 



3K = ( b 2 &„_ x \ 



9a } \«2 a 3 • • • a «-l a n) ) 



3K_ = / 6, &„_ s \ 



96,1-! \ a l «2 • • • a n-Z a n-i) • 



(3) It is thus seen that in whatever diagonal the independent variable is situated 

 the differential-quotient of K is in general the product of two coaxial minors of K, and 

 that when the independent variables are in the same row the differential- quotients have 

 a factor in common. As a result of this 



( b n-i \ 



3K ^ 3K \a h+1 . . . a n _ x aJ 



3«/i 96a " ( K-i \ ' 



\ a 7i+2 • • • a n-l ®n/ 

 — n 4- *+' 7) 



dk+2 + — . 



a h+3 + • . 



+ ht=i 



(4) If we differentiate dlijda h with respect to another element of the main diagonal, 

 that element must be in one of the two factors of dK/da h , 



( *! ) ( & »-> ) 



\a x a^ ... a h _J , \a h+1 . . . a„_ x aJ . 



and not in the other, so that the result of the second differentiation is in general to give 

 an expression of three factors. Thus if 



X = ( a Py 8 € £ 



\a b c d ef g y 

 we have 



9c.9/ "" \a b) ' \d e) ' 9 • 



When the two elements used as independent variables belong to the minor diagonal, 

 the like consequences ensue : it has to be noted however that if they be consecutive the 

 result is 0, because when we differentiate with respect to b h , neither ^.j nor b h+1 

 appears in the result. 



(5) As each term of the continuant is linear in each of the elements occurring in it, 



